Find the area of an equilateral triangle

[anemone]

Show that the curve $x^3+3xy+y^3=1$ has only one set of three distinct points, $P$, $Q$, and $R$ which are the vertices of an equilateral triangle, and find its area.

 

[MarkFL]

Here is my solution:

Spoiler

The first thing I notice is that there is cyclic symmetry between $x$ and $y$, and so setting $y=x$, we find:

$$2x^3+3x^2-1=(x+1)^2(2x-1)=0$$

Thus, we know the points:

$$(x,y)=(-1,-1),\,\left(\frac{1}{2},\frac{1}{2} \right)$$

are on the given curve. Next, if we begin with the line:

$$y=1-x$$

and cube both sides, we obtain:

$$y^3=1-3x+3x^2-x^3$$

We may arrange this as:

$$x^3+3x(1-x)+y^3=1$$

Since $y=1-x$, we may now write

$$x^3+3xy+y^3=1$$

And since the point $$\left(\frac{1}{2},\frac{1}{2} \right)$$ is on the line $y=1-x$, we know the locus of the given curve is the line $y=1-x$ and the point $(-1,-1)$. Hence, there can only be one set of points on the given curve that are the vertices of any triangle, equilateral or otherwise.

Using the formula for the distance between a point and a line, we find the altitude of the equilateral triangle will be:

$$h=\frac{|(-1)(-1)+1-(-1)|}{\sqrt{(-1)^2+1}}=\frac{3}{\sqrt{2}}$$

Using the Pythagorean theorem, we find that the side lengths of the triangle must be:

$$s=\frac{2}{\sqrt{3}}h=\sqrt{6}$$

And so the area of the triangle is:

$$A=\frac{1}{2}sh=\frac{1}{2}\sqrt{6}\frac{3}{\sqrt{2}}=\frac{3\sqrt{3}}{2}$$

 

[anemone]

Here is my solution:

Spoiler

The first thing I notice is that there is cyclic symmetry between $x$ and $y$, and so setting $y=x$, we find:

$$2x^3+3x^2-1=(x+1)^2(2x-1)=0$$

Thus, we know the points:

$$(x,y)=(-1,-1),\,\left(\frac{1}{2},\frac{1}{2} \right)$$

are on the given curve. Next, if we begin with the line:

$$y=1-x$$

and cube both sides, we obtain:

$$y^3=1-3x+3x^2-x^3$$

We may arrange this as:

$$x^3+3x(1-x)+y^3=1$$

Since $y=1-x$, we may now write

$$x^3+3xy+y^3=1$$

And since the point $$\left(\frac{1}{2},\frac{1}{2} \right)$$ is on the line $y=1-x$, we know the locus of the given curve is the line $y=1-x$ and the point $(-1,-1)$. Hence, there can only be one set of points on the given curve that are the vertices of any triangle, equilateral or otherwise.

Using the formula for the distance between a point and a line, we find the altitude of the equilateral triangle will be:

$$h=\frac{|(-1)(-1)+1-(-1)|}{\sqrt{(-1)^2+1}}=\frac{3}{\sqrt{2}}$$

Using the Pythagorean theorem, we find that the side lengths of the triangle must be:

$$s=\frac{2}{\sqrt{3}}h=\sqrt{6}$$

And so the area of the triangle is:

$$A=\frac{1}{2}sh=\frac{1}{2}\sqrt{6}\frac{3}{\sqrt{2}}=\frac{3\sqrt{3}}{2}$$

Awesome, MarkFL, awesome!(Sun) This is your second time answered to my challenge problem and thank you for participating!

 

[Saitama]

Why both of you posted the same solution?

 

[anemone]

Why both of you posted the same solution?

I am so so sorry for causing the confusion. I was meant to reply to MarkFL by quoting his solution, and then typed beneath it and I accidentally clicked on the "Edit" button(I could edit it since I'm the moderator in the "Challenge Questions and Puzzles" subforum), and when I realized I bungled that I immediately rectified the situation by posting my reply and again, I made another mistake, those solution should be put under quote tag but I didn't. Sorry again.

 

[MarkFL]

I fixed it. (Hug)

 

[anemone]

I fixed it. (Hug)

Thanks, Mark...you are forever my sweetest admin!